Hindawi Publishing CorporationThe Scientific World JournalVolume 2013, Article ID 734387, 5 pageshttp://dx.doi.org/10.1155/2013/734387
Research ArticleGeometric Nonlinear Analysis of Self-Anchored Cable-StayedSuspension Bridges
Wang Hui-Li,1 Tan Yan-Bin,1 Qin Si-Feng,2 and Zhang Zhe1
1 Bridge Engineering Research Institute, Dalian University of Technology, Dalian 116085, China2 Research Center for Numerical Tests on Material Failure, Dalian University, Dalian 116622, China
Correspondence should be addressed to Wang Hui-Li; [email protected]
Received 5 August 2013; Accepted 9 September 2013
Academic Editors: J. Mander, L. Qu, and I. B. Topcu
Geometric nonlinearity of self-anchored cable-stayed suspension bridges is studied in this paper.The repercussion of shrinkage andcreep of concrete, rise-to-span ratio, and girder camber on the system is discussed. A self-anchored cable-stayed suspension bridgewith a main span of 800m is analyzed with linear theory, second-order theory, and nonlinear theory, respectively. In the conditionof various rise-to-span ratios and girder cambers, the moments and displacements of both the girder and the pylon under liveload are acquired. Based on the results it is derived that the second-order theory can be adopted to analyze a self-anchored cable-stayed suspension bridge with a main span of 800m, and the error is less than 6%. The shrinkage and creep of concrete impose aconspicuous impact on the structure. And it outmatches suspension bridges for system stiffness. As the rise-to-span ratio increases,the axial forces of the main cable and the girder decline. The system stiffness rises with the girder camber being employed.
1. Introduction
The self-anchored cable-stayed suspension bridge dates backto the early 19th century [1, 2]. First a mixed structureof cable-stayed and suspension bridge was built in France.After years of trial and effort, it has evolved to Roeblingsystem, Dichinger system, and then the improved Dichingersystem.The suspended part of the self-anchored cable-stayedsuspension bridge is much shorter than that of suspensionbridge with the same overall span; therefore the main cableforce can be reduced in a large extent. What is more,the cantilever length of cable-stayed part can be decreasedgreatly during the construction period, thereby improvingaerodynamic stability of the structure.This bridge system hasbeen proposed for many design projects, such as the Strait ofGibraltar Bridge project and the Lingdingyang Bridge project[3].
Until now the cable-stayed suspension bridge stillremains in the design proposal phase. No large-span cable-stayed suspension bridge has ever been built in the world.And all the design proposals are earth-anchored systems.Both the anchorage and the construction period can be
saved through adopting the self-anchored system. The self-anchored cable-stayed suspension bridge suits the long-span needs well. Long-span cable-stayed bridges, suspensionbridges, and cable-stayed suspension bridges have been dis-cussed a lot, but there are few papers relating to self-anchoredcable-stayed suspension bridges [4]. Based on a self-anchoredcable-stayed suspension bridge with an 800m main span, itsgeometric nonlinearity under live load is studied.
2. The Geometric Nonlinear Characteristicsof the Cooperation System
Three main factors cause the geometric nonlinearity of thecooperation system, including the cable sag, large displace-ments, and the initial internal force.
2.1. The Cable Sag Effect. The cable would sag in the freesuspension state. The sag and the chord length of the cablechange as the internal force alters. A nonlinear relationshipexists between the chord length and the cable force. Themethod of equivalent elasticity modulus can be adopted to
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simulate the sag effect. The well-known Ernst formula forequivalent elasticity modulus is shown as follows:
𝐸eq =𝐸
[1 + 𝑤2𝐿2𝐴𝐸/ (12𝑇3)], (1)
where 𝐸 indicates the elasticity modulus of the cable, 𝐿 thehorizontal projection length of the cable element,𝑤 the cableweight per unit length,𝐴 the cross-sectional area of the cable,and 𝑇 the axial force of the cable element. The cable elementcan be set up as a straight bar with the elastic moduluscorrection.
2.2. The Effect of the Initial Internal Force and Large Displace-ments. The girder and main pylon of the cooperation systembear tremendous pressure. The axial force causes additionalbending moments, which affects the bending stiffness ofcomponents; meanwhile the bending moment alters thelength of structure components, which furthermore affectsthe axial stiffness of components. By introducing the initialstress stiffness matrix [𝐾]
𝐺to simulate the effect of the initial
internal force and stiffness matrix of displacement [𝐾]𝐿to
model the effect of large displacements, the initial tangentstiffness matrix can be attained [5]:
[𝐾] = [𝐾]0 + [𝐾]𝐺 + [𝐾]𝐿, (2)
where [𝐾]0represents linear stiffness matrix.
2.3. Second-Order Analysis Theory. Geometric nonlinearityleads to the failure in the application of superpositionprinciples. The moment and displacement caused by liveload cannot be calculated with influence lines. The responseof the cooperation system under all kinds of loads can beattained with the method described above. However, it istime-consuming, especially for the live load which requiresrepeated iterations. Second-order theory, a simplifiedmethodof approximate calculation, will be discussed as follows.
The girder and pylon of the cooperation system belong tocompression-bending members. A simply planar differentialequation for compression bending beams is shown below [6]:
𝐸𝐼](4) + 𝑁 (𝑥) ](2) = 𝑞 (𝑥) . (3)
The axial force 𝑁(𝑥) of the cooperation system consistsof two parts. The one caused by the dead load is recorded as𝑁𝑔(𝑥); the other one caused by live load is denoted by𝑁
𝑞(𝑥).
Therefore, (3) is a nonlinear differential equation.Dr. Li Guohao solved the second-order nonlinear cate-
naries theory with linear method [7]. By drawing on thisidea, the nonlinear analysis of the cooperation system can besimplified. The proportion of live load to dead load for long-span bridges usually ranges from 10% to 20% [8]. Thereforethe effect of 𝑁
𝑞(𝑥) is negligible. Then (3) can be simplified
into
𝐸𝐼](4) + 𝑁𝑔(𝑥) ](2) = 𝑞 (𝑥) . (4)
Equation (4) is a linear differential equation, whichsatisfies the condition of linear superposition.This simplifiedmethod can be called second-order theory [9].
3. Case Study
The Dalian Gulf Bridge with a main span of 800m, is inthe form of a self-anchored cable-stayed suspension bridge.Its total length is 1326m. The main girder section adoptsstreamlined flat box girder, which is 3.5m high and 34mwide. There are two kinds of main girder for the wholebridge including steel girder and prestressed concrete girder.The steel part is in the middle suspension segment and theprestressed concrete part in the cable-stayed area. The H-shaped pylon is 127m above the main girder. The elevationlayout of the bridge is shown in Figure 1.
3.1. Study on the Nonlinearity of Live Load. Here the non-linearity of live load is analyzed with nonlinear theory,second-order theory, and linear theory, respectively. Sincethe superposition principle becomes inappropriate in thenonlinear analysis, the influence zone method is adopted.Under the automobile load of grade I (China), the momentand displacement envelope diagram of main girder and mainpylon are as shown in Figures 2, 3, 4, and 5. The results ofthese calculations are summarized in Table 1.
(1) The result of linear theory radically differs fromthat of nonlinear theory, while the result of second-order theory is close to that of nonlinear theory. Themaximum relative difference between the results ofsecond-order theory and nonlinear theory is less than6%, which is acceptable for the actual project. Thecalculation of second-order theory is simpler thanthat of nonlinear theory; therefore for the cooperationsystem bridge with a main span of 800m, the second-order theory is feasible to calculate live load response.
(2) The maximum positive moment of the main girderoccurs at the middle, while the maximum nega-tive moment occurs near the junction. Due to thegreat change of stiffness and the enormous nega-tive moment, the junction between steel girder andprestressed concrete girder has to be strengthenedspecifically.
(3) Themaximumdeflection at themiddle of the girder is1.18m, which is about 1/650 of themiddle span length.Therefore, the integral stiffness of the cooperation sys-tem bridge is higher than that of normal suspensionbridges with the same length, due to the fact that thecable-stayed part enhances the global stiffness.
3.2. Study on Concrete Shrinkage and Creep. Shrinkage andcreep of concrete can make girder and pylon shorter, causingthe main cable and stayed-cable sag, so the bending momentand deformation of the girder increase. Based on the BridgeCriterion (China), the concrete shrinkage and creep effectswithin 15 years are analyzed. The results are summarized inTable 2.
The results show that concrete shrinkage and creep havean effect on internal forces and the shape of cooperationsystem. Effective measures should be taken to reduce theinfluence, such as extending the load age of concrete, usingmicroexpansion concrete and so on.
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0.00
26383 180 83180
2631326800
35 × 7 + 15 = 260 35 × 7 + 15 = 26028 × 10 =280
(a) Elevation of bridge/m3.
5
11.25
2% 2%
20.6
34
1.6 0.5 0.5 0.40.35
11.25 2 0.61.60.50.5 0.3531.2
Lane LaneSidewalk Sidewalk
(b) Standard section of steel box beam
2% 2%
3.5
78.6 8.6731.2
11.2520.6
34
1.6 0.5 0.5 0.40.35 11.25 2 0.61.60.50.5 0.3531.2
(c) Standard section of concrete box beam
Figure 1: Arrangement diagram of Dalian Gulf Bridge/m.
Table 1: Nonlinear effect of live load.
Nonlinearity Second-order Relatively deviation% Linear Relatively deviation%Moment at midspan ofgirder/(103 KN⋅m)
3.3. Analysis on the Impact of Rise-to-Span Ratio. The rise-to-span ratio of the main cable is an important parameter forthe cooperation system bridge, which affects both structural
−1.2−1.0−0.8−0.6−0.4−0.2
0.0MidspanJuncturePylonAuxiliary pier
Disp
lace
men
t (m
)
Coordinates X of girder (m)
NonlinearitySecond-order linearLinear
−600 −400 −200 200 400 6000
Figure 3: Envelope of main girder displacement.
stiffness and internal forces. The impact of rise-to-span ratiounder live load is given in Figure 6. For the sake of conve-nience, dimensionless forms are adopted. It is convenient to
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NonlinearitySecond-order linearLinear
−60
−40
−20
0
20
40
60
80
100
120
Coo
rdin
ates
Yof
pyl
on (m
)
−2 −1 0 1 32
Moment (105 KN·m)
Figure 4: Envelope of pylon moment.
make the parameters dimensionless using results of rise-to-span ratio 1/10 as a reference.
Figure 6 shows that global stiffness increases as the rise-to-span ratio rises, which is the same as the cooperationsystem bridge but opposite to earth-anchored suspensionbridges [10]. The axial forces of the girder and the maincable descend as the rise-to-span ratio increases. The axialforce of the girder is extremely sensitive to the rise-to-spanratio. The axial forces of the girder and the main cableare enormous, which affects the cross-sectional areas of thegirder and the main cable. So the smaller rise-to-span ratio isnot recommended for the cooperation system bridge.
Figure 6 also shows that horizontal displacements ofthe main girder descend as the rise-to-span ratio increasesbecause the axial force of the girder descends as the rise-to-span ratio increases. As the rise-to-span ratio increases, themoment at the pylon root rises a little.
3.4. Analysis on Camber of the Main Girder. Because of thehigh level of component force of the main cable at bothends of the main girder, the camber of the main girderwill induce an additional negative bending moment for themain girder and further increase the main girder’s momentdue to P-Δ effect. The moment under dead load can beadjusted through the cable tension adjustment; therefore itis only necessary to analyze the main girder camber’s effectunder live load. Table 3 presents the main girder’s bendingmoments and deflections under live load with both 0m and2.9m cambers at the midspan of the main girder. The resultsindicate that setting up a girder camber can effectively reducethe bending moment of the main girder and improve thestructural stiffness.
NonlinearitySecond-order linearLinear
Coo
rdin
ates
Yof
pyl
on (m
)
−60
−40
−20
0
20
40
60
80
100
120
Displacement (m)−0.03 0.00 0.03 0.06 0.09 0.12
Figure 5: Envelope of pylon displacement (positive directionpointing to mid-span).
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Rela
tive r
atio
f/l
Midspan moment Pylon root moment Max. axis force of main cable Max. axis force of main girder Midspan deflection Horizontal displacement of main girder
Geometric nonlinear factors of the cooperation systembridgeare discussed in this paper. Based on this, a cooperationsystem bridge with an 800m main span is analyzed. And thefollowing conclusions are reached.
(1) The error is less than 6% using a simple second-orderapproximation theory to calculate live load response
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Table 2: Effect of shrinkage and creep of concrete.
Position Moment/(KN⋅m) Displacement/(mm)Girder
Midspan −358 −36.6Supporting point atpylon −11700 −1.0
Pylon 36150 27.2The moment of pylon takes place at the root. The displacement of pylontakes place at top and the positive direction points to midspan. The positivedirection of girder displacement points to upward side.
Table 3: Effect of camber of main girder.
0 camber 2.9m camber Relativelydeviation%
Midspanmoment/(103 KN⋅m) 101.19 99.18 −1.99
Midspandeflection/mm 1186 1151 −2.95
of a cooperation system bridge with an 800m mainspan.
(2) The stiffness of the junction between the cable-stayed segment and the suspension area varies. Greatinternal forces occur easily under live load, so it isnecessary to strengthen the junction.
(3) Concrete shrinkage and creep have a conspicuousimpact on the internal force and deformation of thestructure. It is necessary to take measures to alleviatethe influence.
(4) Global stiffness increases with the rise-to-span ratioascending and the axial forces of the girder andthe main cable descend as the rise-to-span ratiorises. Therefore the small rise-to-span ratios are notrecommended for the cooperation system bridge.
(5) Setting a girder camber can improve the integralstiffness of the cooperation system bridge.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
This work is funded by National Natural Science Foundation(51008047, 51108052) and West Transportation ConstructionProjects Foundation of Ministry of Communications, China(2006 318 823 50).
References
[1] N. J. Gimsing, Cable Supported Bridges: Analysis and DesignConcept, John Wiley & Sons, Chichester, UK, 1997.
[2] B. H. Wang, “Cable-stayed suspension bridges,” Journal ofLiaoning Provincial College of Communications, vol. 2, no. 3, pp.1–6, 2000.
[3] R. C. Xiao, L. J. Jia, and E. L. Xue, “Research on the designof cable-stayed suspension bridges,” China Civil EngineeringJournal, vol. 33, no. 5, pp. 46–51, 2000.
[4] H. L. Wang, Structure Properties Analysis and ExperimentStudy of Self-AnchoredCable-Stayed SuspensionBridge [Doctoraldissertation], Dalian University of Technology, Dalian, China,2007.
[5] C. X. Li and G. Y. Xia, “Geometric nonlinear analysis of longspan bridge,” in The Calculation Theory of Long Span BridgeStructure, pp. 88–113, China Communications Press, Beijing,China, 2002.
[6] X. F. Sun, X. S. Fang, and T. L. Guan, “The beam deflections,” inMechanics of Materials, pp. 269–309, China Higher EducationPress, Beijing, China, 1994.
[7] G. H. Li, “Utility calculation of suspension bridges with second-order theory,” in Study of Bridge and StructureTheory, ShanghaiScience and Technology Publishing House, Shanghai, China,1983.
[8] H. F. Xiang, “The calculation theory of cable-stayed bridges,”in Higher Theory of Bridge Structure, pp. 282–301, ChinaCommunications Press, Beijing, China, 2002.
[9] Y. R. Pan, “Suspension bridge analysis under external loads,”in Nonlinear Analysis Theory and Method of Suspension BridgeStructure, pp. 69–89, China Communications Press, Beijing,China, 2004.
[10] W. L. Qiu Zhang Z and C. L. Hang, “Mechanical properties ofself-anchored concrete suspension bridge,” Journal of HarbinUniversity of Civil Engineering and Architecture, vol. 35, no. 11,pp. 1388–1391, 2003.
